Siegel-Veech constants and Masur-Veech volumes: Connection to Intersection theory

• Applicant: Professor Dr. Martin Möller
• Subject Area: Mathematics
• Term: from 2015 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 275218485

• Final Report Year: 2025

Final Report Abstract

Siegel-Veech constants measure the asymptotic number of closed trajectories up to a given length bound on a polygonal billiard table and thus form a fundamental invariant of the billiard dynamical system. Via an unfolding process they can be interpreted as a counting problem on a flat surface, a Riemann surface with a holomorphic one-form. In fact, Siegel-Veech constants can be computed by understanding the geometry of the moduli space of flat surfaces, notably intersection numbers on them. This mirrors the case of the moduli space of curves, where intersection numbers are well-studied, at least since Witten’s conjectures. There is a natural action of the group GL2 (R) on flat surfaces and Siegel-Veech constants depend only on the GL2 (R)-orbit closure. These closures are so-called linear submanifolds, by fundamental work of Eskin, Mirzakhani, Mohammadi. In this project we proved a conjecture relating Siegel-Veech constants and ratios of intersection numbers on linear manifolds under the additional condition that the linear manifold has ’no relative period’. This condition is satisfied e.g. for most of the exceptional examples recently discovered by Eskin, McMullen, Mukamel, Wright. We also compute the Chern classes of the cotangent bundle of linear manifolds, preparing the ground for studying their birational geometry. Moreover, the project provided several steps towards determining the cohomology of the moduli space of flat surfaces, notably towards determining its boundary complex.

Publications

• An Algorithm to Compute the Fundamental Classes of Spin Components of Strata of Differentials. International Mathematics Research Notices, 2024(6), 4893-4962.
  Wong, Yiu Man
  
• Masur–Veech volumes and intersection theory: The principal strata of quadratic differentials. Duke Mathematical Journal, 172(9).
  Chen, Dawei; Möller, Martin & Sauvaget, Adrien
  
• Geometry of strata of differentials on Riemann surfaces, PhD thesis, Goethe Universiät Frankfurt, 2024, 144 pages.
  Y. M. Wong
  
• Linear submanifolds and strata of k-differentials : invariants and applications, PhD thesis, Goethe Universiät Frankfurt, 2024, 141 pages.
  J. Schwab
  
• Chern classes of linear submanifolds with application to spaces of k-differentials and ball quotients. Commentarii Mathematici Helvetici, 100(4), 745-809.
  Costantini, Matteo; Möller, Martin & Schwab, Johannes
  
• Rational cohomology of M4,1. Mathematische Nachrichten, 298(3), 1041-1061.
  Wong, Yiu Man & Zheng, Angelina